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Chapter 1: Systems of Linear

Equations and Matrices
Section 1.2: Gaussian Elimination (and Matrices)

Section 1.2: Matrices and Elementary Row Operations

 As the size of the linear system increases, the complexity of the algebra involved in finding solutions also increases.
 Need to simplify notations => MATRICES

Definition: An × matrix is a rectangular array of numbers with rows and

columns.
The numbers in this array are called the elements (entries) of the matrix. The element of a
matrix in row and column is denoted

Row index Column index

Example
1 0 −2
=
9 3 5
has 2 rows and 3 columns ⇒ is a 2 × 3 matrix.

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Examples of Matrices:
1 0 −2
1) = is a rectangular matrix (2 × 3)
9 3 5
1 2
2) = is a square matrix (2 × 2)
3 0
A square matrix is a matrix that has the same number of columns as the number of rows i. e. ( × , × ,…)
1 5 3
3) = 0 0 2 is an upper triangular square matrix.
0 0 4
A matrix is an upper triangular matrix provided that = 0 for all > ,i.e. all entries below the main diagonal are zero.

1 0 0
4) = 2 −1 0 is an lower triangular square matrix.
−3 0 2
A matrix is a lower triangular matrix provided that = 0 for all > i ,i.e. all entries above the main diagonal are zero.

The main diagonal of a matrix is all entries having the same row index as column index i.e. for all .
5) = 1 −1 2 is a row matrix or a row vector (1 × 3).

6) = 2 is a column matrix or a column vector (2 × 1).

0.5

+ + … + =
+ + … + =
 An × linear system
⋮ ⋮ ⋮ … ⋮ ⋮
+ + ⋯ + =
can be abbreviated in the form of an × ( + 1) augmented matrix

…
…
⋮ ⋮ ⋮ … ⋮
⋯

…
…
The coefficient matrix:
⋮ ⋮ ⋮ ⋮
⋯

Example
3 2 −1 0 2
3 +2 − =2
1 −15 0 1 3
− 15 + =3 Note: We use 0 to record a missing term
−2 44 5 30
−2 + 44 + 5 +3 =0

3 × 4 linear system 3 × 5 augmented matrix

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The operations on a linear system that produce an equivalent The elementary row operations on a matrix that produce an
linear system: equivalent matrix:

1. Interchanging any two equations. 1. Interchanging any two rows.

2. Multiplying any equation by a nonzero constant. 2. Multiplying any row by a nonzero constant.

3. Adding a multiple of one equation to another. 3. Adding a multiple of one row to another.

Example
-2E +E
1 2
2E2+E3
E1 + + =2 + + =2 +2 =3
E2 2 + 3 + = 3 -E1+E3 − = −1 -E2+E1 − = −1
E3 − −2 = −6 −2 − 3 = −8 −5 = −10

-2R1+R2 2R2+R3
R1 1 1 1 2 1 1 1 2 1 0 2 3 +2 =3
R2 2 3
-R1+R3 0 1 −1 −1 -R2+R1 0 1 −1 −1 ~ − = −1
1 3
R3 1 −1 −2 −6 0 −2 −3 −8 0 0 −5 −10 −5 = −10

By back substitution: −5 = −10 ⇒ =2

− = −1 ⇒ = −1 + =1 So = (−1,1,2) is a unique solution
+2 =3 ⇒ =3−2 = −1

Definition: An × matrix is in row echelon form if

1. Every row of all 0 entries (if any) is placed at the bottom of the matrix.
2. In any two consecutive nonzero rows, the leftmost nonzero entry (leading entry or
pivot) of the lower row must be further to the right than the pivot of the upper row.
(i.e. The pivot in row ( + 1) is to the right of the pivot in row .)

The matrix is in reduced row echelon form if, in addition,

3. Every pivot is 1.
4. Each column that contains a pivot has all other entries 0.

Example: For the following matrices, determine whether they are in REF or/and in RREF.
0 2 4 6 1 5 1 0 0 4 7 1 0 4 7 1 2 0 8
0 0 3 8 0 1 0 0 1 2 8 0 1 2 8 0 0 1 6
0 1 5 7 0 0 1 0 0 1 5 0 0 0 0 0 0 0 1
0 0 0 0

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If in REF, a row of the form 0 0 … 0 , with

≠ 0 is obtained, then the system is
inconsistent

If the last column of the augmented matrix contains a

Linear Augmented pivot the system is inconsistent!!
REF
System Matrix

Otherwise, the system is consistent and the

Gaussian Elimination
pivots correspond to dependent variables.
The non-leading variables are the “free”
variables and they are labeled by new
variables (s,t, …) called “parameters”
If number of pivots = number of variables, the
consistent system has a unique solution!

Example: The following matrices are the RREF of the augmented matrices corresponding to some linear systems.
Determine whether the systems are consistent or inconsistent. When consistent, find the solution set.

1 0 0 2 1 1 00
1) 0 1 0 3 2) 0 0 10
0 0 1 −5 0 0 01

1 0 8 04
3) 0 1 0 02
0 0 0 14

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5 +3 + 7− = +5
Example: Determine the values of for which the linear system −4 + 1− =1 has:
−2 =3
a) a unique solution
b) no solution
c) Infinitely many solutions

Example: Find the solution sets of the following linear systems by Gaussian elimination followed by back
substitution.

+3 −2 +4 =5
1)
2 −2 + +5 =8

2 − 3 = −2
2) 2 + =1
4 +2 = 2

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1 2 3
Example: Find the RREF of the matrix 2 4 2
1 0 0

 The process of transforming a matrix to reduced row echelon form is called “Gauss-Jordan elimination”.
 The reduced row echelon form of a matrix is unique
 Row echelon forms of matrices are not unique
 Two matrices and B are called row equivalent if either can be obtained from the other by elementary row operations.
Example: Find the solution set of the following linear systems by Gaussian-Jordan elimination.
− 3 = −3
+ 2 = −3

Example: Determine according to the values of and the number of solutions of:
6 − 15 =
i.
4 − 10 =

+ = 1
ii.
+ =4

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0
Example: Given the matrix = 0 . Find the values of , and so that is:
0 +
i. Upper triangular
ii. In REF
iii. In RREF

Homogeneous Linear Systems

Definition: A linear system of m linear equations in n variables is called a homogeneous linear system, if all the
constants on the right side are 0:
a11x1 + a12x2 + · · · + a1nxn = 0
a21x1 + a22x2 + · · · + a2nxn = 0
a31x1 + a32x2 + · · · + a3nxn = 0
...
...
...
...
am1x1 + am2x2 + · · · +amnxn = 0
+ =0
Example: Consider a 2 × 2 homogeneous linear system , where , not both zero and , not both
+ =0
zero.

Each equation represents a line in the −plane passing through the origin. So, they must intersect at least in the origin.

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+3 −2 =0
Example: Find the solution set of the homogeneous linear system 2 +6 −5 −2 =0
5 + 10 =0

 A homogeneous linear system always has at least the trivial solution which is = 0, = 0, … , = 0.
Solutions other than the trivial solution are called nontrivial solutions.
 A homogeneous linear system is consistent, it has either a unique solution or infinitely many
solutions.
 If the REF of the augmented matrix of an × homogeneous linear system has pivots, then the
homogeneous linear system has a unique solutions. Otherwise, it has infinitely many solutions and
Number of free variables= − number of pivots.

 An × homogeneous linear system where < , has infinitely many solutions.

Free Variable Theorem for Homogeneous Systems

If a homogeneous linear system has n unknowns, and if the reduced row echelon form of its augmented matrix has r
nonzero rows, then the system has n − r free variables.

A homogeneous linear system with more unknowns than equations has infinitely many solutions.

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Summary
 An m × n linear system may be written in the form of an m × (n+1) augmented matrix.
 To solve the system, the augmented matrix is reduced to an equivalent matrix in row echelon form.
 The elementary row operations used to reduce the matrix to an equivalent matrix without altering the
solution are:
1. Interchanging any two rows
2. Multiplying any row by a nonzero constant
3. Adding a scalar multiple of one row to another row

 If in REF, a row of the form 0 0 … 0 1 is obtained, then the system is inconsistent

 If the system is consistent and the number of pivots equals the number of variables, there is a unique
solution.
 If the system is consistent and the number of pivots is less than the number of variables, there are
infinitely many solutions and
number of free variables (or parameters) = total number of variables – number of pivots.

 If the augmented matrix of an n × n linear system is row-reduced to REF and the coefficient matrix
has no rows of zeros, then the linear system has a unique solution.